Pseudogroups and their étale groupoids
Pseudogroups and their étale groupoids
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伪群及其群形
DOI：
10.1016/j.aim.2013.04.022
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发表时间：
2011
影响因子：
1.7
通讯作者：
D. Lenz
中科院分区：
文献类型：
--
作者：
M. Lawson;D. Lenz
伪群是完全的无限分配逆幺半群。这样的逆幺半群与经典伪变换群的关系就像框架与拓扑空间的关系一样。本文的目标是发展理论的伪群动机的应用群论，C-代数和非周期平铺。我们的出发点是一类伪群和一类étale群胚之间的一个连接，由此我们能够建立空间伪群和sober étale群胚之间的对偶。作为这个对偶的推论，我们推导出一个非交换版本的斯通对偶涉及我们所谓的布尔逆半群和布尔代数群胚，以及一个广义的分配逆半群的对偶。非交换Stone对偶在C-代数理论中有重要的应用：它是构造Cuntz代数和Cuntz-Krieger代数的基础，在Cuntz代数的情况下，它也可以用来构造Thompson群。然后，我们定义覆盖逆半群和由此产生的介绍伪群。作为应用程序，我们表明，帕特森的普遍广群是一个布尔化的例子，并调和埃克塞尔最近的工作理论的紧映射与工作的第二作者。
A pseudogroup is a complete infinitely distributive inverse monoid. Such inverse monoids bear the same relationship to classical pseudogroups of transformations as frames do to topological spaces. The goal of this paper is to develop the theory of pseudogroups motivated by applications to group theory, C∗-algebras and aperiodic tilings. Our starting point is an adjunction between a category of pseudogroups and a category of étale groupoids from which we are able to set up a duality between spatial pseudogroups and sober étale groupoids. As a corollary to this duality, we deduce a non-commutative version of Stone duality involving what we call boolean inverse semigroups and boolean étale groupoids, as well as a generalization of this duality to distributive inverse semigroups. Non-commutative Stone duality has important applications in the theory of C∗-algebras: it is the basis for the construction of Cuntz and Cuntz–Krieger algebras and in the case of the Cuntz algebras it can also be used to construct the Thompson groups. We then define coverages on inverse semigroups and the resulting presentations of pseudogroups. As applications, we show that Paterson’s universal groupoid is an example of a booleanization, and reconcile Exel’s recent work on the theory of tight maps with the work of the second author.